cartesian closed category

Let C be a category with binary products and let Y and Z be objects of C. The exponential object ZY can be defined as a universal morphism from the functor –×Y to Z. (The functor –×Y from C to C maps objects X to X×Y and morphisms φ to φ×idY). Explicitly, the definition is as follows. An object ZY, together with a morphism is an exponential object if for any object X and morphism g : (X×Y) → ZSee also:

Exponential object, Exponential object - Definition, Exponential object - Examples

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Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "generalized abstract nonsense". Categories appear in most branches of mathematics, in some areas of theoretical computer science and mathematical physics, and have been a unifying notion. Categories were first introduced by Samuel Eilenberg and Saunders Ma ...

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• Category theory - Background
• Category theory - Historical notes
• Category theory - Categories objects and morphisms
• Category theory - Some properties of morphisms
• Category theory - Functors
• Category theory - Natural transformations and isomorphisms
• Category theory - Universal constructions limits and colimits
• Category theory - Equivalent categories
• Category theory - Further concepts and results
• Category theory - Higher-dimensional categories

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In the category of sets, the exponential object ZY is the set of all functions from Y to Z. The map is just the evaluation map which sends the pair (f, y) to f(y). For any map the map is the curried form of g: In the category of topological spaces, the exponential object ZY exists provided that Y< ...

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Exponential object, Exponential object - Definition, Exponential object - Examples

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Dana Scott took up a post as Professor of Mathematical Logic, in the Philosophy Faculty of Oxford University in 1972. Dana Scott - Semantics of programming languages. This period saw Scott working close to Christopher Strachey, and the two managed, despite intense administrative pressures, to oversee a great deal of fundamental work on providing a mathematical foundation for the semantics of programming languages, the work for which Scott is best known. Together their work constitutes the Scott-Strachey ap ...

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Dana Scott, Dana Scott - Early career, Dana Scott - University of California Berkeley 1960–1963, Dana Scott - Stanford Amsterdam and Princeton 1963–1972, Dana Scott - Oxford University 1972–1981, Dana Scott - Semantics of programming languages, Dana Scott - Carnegie Mellon University 1981–2003

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Function space - Functional analysis. Schwartz space of smooth functions of rapid decrease and its dual, tempered distributions Lp space κ(R) continuous compact support with uniform norm C(R) bounded continuous (Bounded_function) C∞(R) functions which vanish at infinity C∞(R) Smooth_functions C∞0 smooth compact support uniform norm (and the one with ...

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Function space, Function space - List of function spaces, Function space - Functional analysis

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Given any topological space X we can define a (possibly) finer topology on X which is compactly generated. Let {Kα} denote the family of compact subsets of X. We define the new topology on X by declaring a subset A to be closed iff A ∩ Kα is closed in Kα for each α. Denote this new space by Xc. One can show that the compact subsets of Xc and X coincide and the induced topologies are the same. It fol ...

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Compactly generated space, Compactly generated space - Properties

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Suppose C is a small category (i.e. the objects form a set rather than a proper class) and D is an arbitrary category. The category of functors from C to D, written as Funct(C,D) or DC, has as objects the covariant functors from C to D, and as morphisms the natural transformations between such functors. Note that natural transformations can be composed: if μ(X) : F(X) → G(X) is a natural transformation from the functor ...

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Functor category, Functor category - Definition, Functor category - Examples, Functor category - Facts

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One can show that a functor F : C -> D yields an equivalence of categories if and only if has all of the following three properties: full, i.e. for any two objects c1 and c2 of C, the map MorC(c1,c2) -> MorD(Fc1,Fc2) induced by F is surjective; faithful, i.e. for any two objects c1 and c_{See also:}

Equivalence of categories, Equivalence of categories - Definition, Equivalence of categories - Equivalent characterizations, Equivalence of categories - Examples, Equivalence of categories - Properties

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The study of categories is an attempt to capture what is commonly found in various classes of related mathematical structures. Consider the following example. The class Grp of groups consists of all objects having a "group structure". More precisely, Grp consists of all sets G endowed with a binary operation satisfying a certain set of axioms. One can proceed to prove theorems about groups by making logical deductions from the set of axioms. For example, it is immediately proved from the axioms t ...

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Category theory, Category theory - Background, Category theory - Historical notes, Category theory - Categories objects and morphisms, Category theory - Some properties of morphisms, Category theory - Functors, Category theory - Natural transformations and isomorphisms, Category theory - Universal constructions limits and colimits, Category theory - Equivalent categories, Category theory - Further concepts and results, Category theory - Higher-dimensional categories

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Main articles: universal property, limit (category theory) Using the language of category theory, many areas of mathematical study can be cast into appropriate categories, such as the categories of all sets, groups, topologies, and so on. These categories surely have some objects that are "special" in a certain way, such as the empty set or the product of two topologies. Yet, in the definition of a category, objects are considered to be atomic; i.e. we do not know, whether an object A is a set, a topology, or any other a ...

See also:

Category theory, Category theory - Background, Category theory - Historical notes, Category theory - Categories, objects, and morphisms, Category theory - Some properties of morphisms, Category theory - Functors, Category theory - Natural transformations and isomorphisms, Category theory - Universal constructions, limits, and colimits, Category theory - Equivalent categories, Category theory - Further concepts and results, Category theory - Higher-dimensional categories

Read more here: » Category theory: Encyclopedia II - Category theory - Universal constructions, limits, and colimits

Main articles: category, morphism A category C consists of a class ob(C) of objects: a class hom(C) of morphisms. Each morphism f has a unique source object a and target object b. We write f: a → b, and we say "f is a morphism from a to b". We write hom(a, b) [or Hom(a, b), or homC(a, b)] to denote the hom-class of all morphisms from < ...

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Category theory, Category theory - Background, Category theory - Historical notes, Category theory - Categories, objects, and morphisms, Category theory - Some properties of morphisms, Category theory - Functors, Category theory - Natural transformations and isomorphisms, Category theory - Universal constructions, limits, and colimits, Category theory - Equivalent categories, Category theory - Further concepts and results, Category theory - Higher-dimensional categories

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He received his Bachelor's degree from the University of California, Berkeley in 1954. He wrote his Ph.D. thesis on Convergent Sequences of Complete Theories under the supervision of Alonzo Church while at Princeton, and defended his thesis in 1958. After completing his PhD studies, he moved to the University of Chicago, working as an instructor there until 1960. In 1959, he published a joint paper with Michael O. Rabin, a colleague from Princeton, entitled Finite Automata and Their Decision Problem, which introdu ...

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Dana Scott, Dana Scott - Early career, Dana Scott - University of California Berkeley 1960–1963, Dana Scott - Stanford Amsterdam and Princeton 1963–1972, Dana Scott - Oxford University 1972–1981, Dana Scott - Semantics of programming languages, Dana Scott - Carnegie Mellon University 1981–2003

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Scott took up a post as Assistant Professor of Mathematics, at the University of California, Berkeley, the university of Alfred Tarski, and involved himself with classical issues in mathematical logic, especially set theory and Tarskian model theory. During this period he started supervising PhD students, such as James Halpern (Contributions to the Study of the Independence of the Axiom of Choice), and Edgar Lopez-Escobar (Infinitely Long Formulas with Countable Quantifier Degrees). Scott's work as res ...

See also:

Dana Scott, Dana Scott - Early career, Dana Scott - University of California Berkeley 1960–1963, Dana Scott - Stanford Amsterdam and Princeton 1963–1972, Dana Scott - Oxford University 1972–1981, Dana Scott - Semantics of programming languages, Dana Scott - Carnegie Mellon University 1981–2003

Read more here: » Dana Scott: Encyclopedia II - Dana Scott - University of California Berkeley 1960–1963

Categories, functors and natural transformations were introduced by Samuel Eilenberg and Saunders Mac Lane in 1945. Initially, the notions were applied in topology, especially algebraic topology, as an important part of the transition from homology (an intuitive and geometric concept) to homology theory, an axiomatic approach. It has been claimed, for example by or on behalf of Stanislaw Ulam, that comparable ideas were current in the late 1930s in the Polish school. These ideas were in some ways a continuation of the contributions of Emmy N ...

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Category theory, Category theory - Background, Category theory - Historical notes, Category theory - Categories objects and morphisms, Category theory - Some properties of morphisms, Category theory - Functors, Category theory - Natural transformations and isomorphisms, Category theory - Universal constructions limits and colimits, Category theory - Equivalent categories, Category theory - Further concepts and results, Category theory - Higher-dimensional categories

Read more here: » Category theory: Encyclopedia II - Category theory - Historical notes

Main article: natural transformation A natural transformation is a relation between two functors. Functors often describe "natural constructions" and natural transformations then describe "natural homomorphisms" between two such constructions. Sometimes two quite different constructions yield "the same" result; this is expressed by a natural isomorphism between the two functors. If F and G are (covariant) functors between the categories C and D, then a natural transformation from FSee also:

Category theory, Category theory - Background, Category theory - Historical notes, Category theory - Categories objects and morphisms, Category theory - Some properties of morphisms, Category theory - Functors, Category theory - Natural transformations and isomorphisms, Category theory - Universal constructions limits and colimits, Category theory - Equivalent categories, Category theory - Further concepts and results, Category theory - Higher-dimensional categories

Read more here: » Category theory: Encyclopedia II - Category theory - Natural transformations and isomorphisms

Main articles: category, morphism A category C consists of a class ob(C) of objects: a class hom(C) of morphisms. Each morphism f has a unique source object a and target object b. We write f: a → b, and we say "f is a morphism from a to b". We write hom(a, b) [or Hom(a, b), or homC(a, b)] to denote the hom-class of all morphisms from < ...

See also:

Category theory, Category theory - Background, Category theory - Historical notes, Category theory - Categories objects and morphisms, Category theory - Some properties of morphisms, Category theory - Functors, Category theory - Natural transformations and isomorphisms, Category theory - Universal constructions limits and colimits, Category theory - Equivalent categories, Category theory - Further concepts and results, Category theory - Higher-dimensional categories

Read more here: » Category theory: Encyclopedia II - Category theory - Categories objects and morphisms

Categories, functors and natural transformations were introduced by Samuel Eilenberg and Saunders Mac Lane in 1945. Initially, the notions were applied in topology, especially algebraic topology, as an important part of the transition from homology (an intuitive and geometric concept) to homology theory, an axiomatic approach. It has been claimed, for example by or on behalf of Stanislaw Ulam, that comparable ideas were current in the later 1930s in the Polish school. These ideas were in some ways a continuation of the contributions of Emmy ...

See also:

Category theory, Category theory - Background, Category theory - Historical notes, Category theory - Categories objects and morphisms, Category theory - Some properties of morphisms, Category theory - Functors, Category theory - Natural transformations and isomorphisms, Category theory - Universal constructions limits and colimits, Category theory - Equivalent categories, Category theory - Further concepts and results, Category theory - Higher-dimensional categories

Read more here: » Category theory: Encyclopedia II - Category theory - Historical notes

Main article: functor Functors are structure-preserving maps between categories. They can be thought of as morphisms in the category of all (small) categories. A (covariant) functor F from the category C to the category D associates to each object x in C an object F(x) in D; associates to each morphism f : x → y a morphism F(f) : F ...

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Category theory, Category theory - Background, Category theory - Historical notes, Category theory - Categories objects and morphisms, Category theory - Some properties of morphisms, Category theory - Functors, Category theory - Natural transformations and isomorphisms, Category theory - Universal constructions limits and colimits, Category theory - Equivalent categories, Category theory - Further concepts and results, Category theory - Higher-dimensional categories

Read more here: » Category theory: Encyclopedia II - Category theory - Functors

Main articles: universal property, limit (category theory) Using the language of category theory, many areas of mathematical study can be cast into appropriate categories, such as the categories of all sets, groups, topologies, and so on. These categories surely have some objects that are "special" in a certain way, such as the empty set or the product of two topologies. Yet, in the definition of a category, objects are considered to be atomic; i.e. we do not know, whether an object A is a set, a topology, or any other a ...

See also:

Category theory, Category theory - Background, Category theory - Historical notes, Category theory - Categories objects and morphisms, Category theory - Some properties of morphisms, Category theory - Functors, Category theory - Natural transformations and isomorphisms, Category theory - Universal constructions limits and colimits, Category theory - Equivalent categories, Category theory - Further concepts and results, Category theory - Higher-dimensional categories

Read more here: » Category theory: Encyclopedia II - Category theory - Universal constructions limits and colimits

Many of the above concepts, especially equivalence of categories, adjoint functor pairs, and functor categories, can be situated into the context of higher-dimensional categories. Briefly, if we consider a morphism between two objects as a "process taking us from one object to another", then higher-dimensional categories allow us to profitably generalise this by considering "higher-dimensional processes". For example, a (strict) 2-category is a category together with "morphisms between morphisms", i.e. processes which allow us ...

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Category theory, Category theory - Background, Category theory - Historical notes, Category theory - Categories objects and morphisms, Category theory - Some properties of morphisms, Category theory - Functors, Category theory - Natural transformations and isomorphisms, Category theory - Universal constructions limits and colimits, Category theory - Equivalent categories, Category theory - Further concepts and results, Category theory - Higher-dimensional categories

Read more here: » Category theory: Encyclopedia II - Category theory - Higher-dimensional categories